doc-src/TutorialI/Ifexpr/Ifexpr.thy

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+(*<*)
+theory Ifexpr = Main:;
+(*>*)
+
+text{*
+\subsubsection{How can we model boolean expressions?}
+
+We want to represent boolean expressions built up from variables and
+constants by negation and conjunction. The following datatype serves exactly
+that purpose:
+*}
+
+datatype boolex = Const bool | Var nat | Neg boolex
+                | And boolex boolex;
+
+text{*\noindent
+The two constants are represented by \isa{Const~True} and
+\isa{Const~False}. Variables are represented by terms of the form
+\isa{Var~$n$}, where $n$ is a natural number (type \isa{nat}).
+For example, the formula $P@0 \land \neg P@1$ is represented by the term
+\isa{And~(Var~0)~(Neg(Var~1))}.
+
+\subsubsection{What is the value of a boolean expression?}
+
+The value of a boolean expression depends on the value of its variables.
+Hence the function \isa{value} takes an additional parameter, an {\em
+  environment} of type \isa{nat \isasymFun\ bool}, which maps variables to
+their values:
+*}
+
+consts value :: "boolex \\<Rightarrow> (nat \\<Rightarrow> bool) \\<Rightarrow> bool";
+primrec
+"value (Const b) env = b"
+"value (Var x)   env = env x"
+"value (Neg b)   env = (\\<not> value b env)"
+"value (And b c) env = (value b env \\<and> value c env)";
+
+text{*\noindent
+\subsubsection{If-expressions}
+
+An alternative and often more efficient (because in a certain sense
+canonical) representation are so-called \emph{If-expressions} built up
+from constants (\isa{CIF}), variables (\isa{VIF}) and conditionals
+(\isa{IF}):
+*}
+
+datatype ifex = CIF bool | VIF nat | IF ifex ifex ifex;
+
+text{*\noindent
+The evaluation if If-expressions proceeds as for \isa{boolex}:
+*}
+
+consts valif :: "ifex \\<Rightarrow> (nat \\<Rightarrow> bool) \\<Rightarrow> bool";
+primrec
+"valif (CIF b)    env = b"
+"valif (VIF x)    env = env x"
+"valif (IF b t e) env = (if valif b env then valif t env
+                                        else valif e env)";
+
+text{*
+\subsubsection{Transformation into and of If-expressions}
+
+The type \isa{boolex} is close to the customary representation of logical
+formulae, whereas \isa{ifex} is designed for efficiency. Thus we need to
+translate from \isa{boolex} into \isa{ifex}:
+*}
+
+consts bool2if :: "boolex \\<Rightarrow> ifex";
+primrec
+"bool2if (Const b) = CIF b"
+"bool2if (Var x)   = VIF x"
+"bool2if (Neg b)   = IF (bool2if b) (CIF False) (CIF True)"
+"bool2if (And b c) = IF (bool2if b) (bool2if c) (CIF False)";
+
+text{*\noindent
+At last, we have something we can verify: that \isa{bool2if} preserves the
+value of its argument:
+*}
+
+lemma "valif (bool2if b) env = value b env";
+
+txt{*\noindent
+The proof is canonical:
+*}
+
+apply(induct_tac b);
+apply(auto).;
+
+text{*\noindent
+In fact, all proofs in this case study look exactly like this. Hence we do
+not show them below.
+
+More interesting is the transformation of If-expressions into a normal form
+where the first argument of \isa{IF} cannot be another \isa{IF} but
+must be a constant or variable. Such a normal form can be computed by
+repeatedly replacing a subterm of the form \isa{IF~(IF~b~x~y)~z~u} by
+\isa{IF b (IF x z u) (IF y z u)}, which has the same value. The following
+primitive recursive functions perform this task:
+*}
+
+consts normif :: "ifex \\<Rightarrow> ifex \\<Rightarrow> ifex \\<Rightarrow> ifex";
+primrec
+"normif (CIF b)    t e = IF (CIF b) t e"
+"normif (VIF x)    t e = IF (VIF x) t e"
+"normif (IF b t e) u f = normif b (normif t u f) (normif e u f)";
+
+consts norm :: "ifex \\<Rightarrow> ifex";
+primrec
+"norm (CIF b)    = CIF b"
+"norm (VIF x)    = VIF x"
+"norm (IF b t e) = normif b (norm t) (norm e)";
+
+text{*\noindent
+Their interplay is a bit tricky, and we leave it to the reader to develop an
+intuitive understanding. Fortunately, Isabelle can help us to verify that the
+transformation preserves the value of the expression:
+*}
+
+theorem "valif (norm b) env = valif b env";(*<*)oops;(*>*)
+
+text{*\noindent
+The proof is canonical, provided we first show the following simplification
+lemma (which also helps to understand what \isa{normif} does):
+*}
+
+lemma [simp]:
+  "\\<forall>t e. valif (normif b t e) env = valif (IF b t e) env";
+(*<*)
+apply(induct_tac b);
+apply(auto).;
+
+theorem "valif (norm b) env = valif b env";
+apply(induct_tac b);
+apply(auto).;
+(*>*)
+text{*\noindent
+Note that the lemma does not have a name, but is implicitly used in the proof
+of the theorem shown above because of the \isa{[simp]} attribute.
+
+But how can we be sure that \isa{norm} really produces a normal form in
+the above sense? We define a function that tests If-expressions for normality
+*}
+
+consts normal :: "ifex \\<Rightarrow> bool";
+primrec
+"normal(CIF b) = True"
+"normal(VIF x) = True"
+"normal(IF b t e) = (normal t \\<and> normal e \\<and>
+     (case b of CIF b \\<Rightarrow> True | VIF x \\<Rightarrow> True | IF x y z \\<Rightarrow> False))";
+
+text{*\noindent
+and prove \isa{normal(norm b)}. Of course, this requires a lemma about
+normality of \isa{normif}:
+*}
+
+lemma [simp]: "\\<forall>t e. normal(normif b t e) = (normal t \\<and> normal e)";(*<*)
+apply(induct_tac b);
+apply(auto).;
+
+theorem "normal(norm b)";
+apply(induct_tac b);
+apply(auto).;
+
+end(*>*)